1. Visa stamp
In your US passport, a visa is a stamp you got from a Thai embassy or consulate and has the word "Visa" on it and information about the type of visa and its issue and expiration dates. Nowadays, in many countries, including the USA, the Thai visas are sent electronically in a PDF file attached to an email message, ie no stamp in your passport.
2. Arrival stamp
The arrival stamp put into your passport by an immigration official when you enter Thailand essentially shows the name of the entry point, the type of visa or visa exemption with which you arrived, the arrival date, and the date until which you are allowed to stay. Basically, this is your permission to stay.